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IC 8871 



Bureau of Mines Information Circular/1982 



A Computer Program for Calculating 
Thermodynamic Properties 
From Spectroscopic Data 



By R. P. Beyer 



UNITED STATES DEPARTMENT OF THE INTERIOR 



Information Circular 8871 

A Computer Program for Calculating 
Thermodynamic Properties 
From Spectroscopic Data 

By R. P. Beyer 




UNITED STATES DEPARTMENT OF THE INTERIOR 
James G. Watt, Secretary 

BUREAU OF MINES 
Robert C. Horton, Director 






^fiO- 



<fl 



n\ 




This publication has been cataloged as follows: 



Beyer, R. P. (Richard P.) 

A computer program for calculating thermodynamic proper- 
ties from spectroscopic data. 

(Information circular ; 8871) 

Bibliography: p. 19. 

Supt. of Docs, no.: 1 28.23:8871. 

1. Thermodynamics— Computer programs. 2. Gases— Thermal 
properties— Computer programs. 3- Gases— Spectra— Computer pro* 
grams. 4. FORTRAN (Computer program language). I. Title. II. 
Series: Information circular (United States. Bureau of Mines) ; 8871. 

TN295.U4 [TJ265] 622s [536'.25] 81-607926 AACR2 



CONTENTS 



Page 

Symbols and terminology ii 

Abstract 1 

Introduction 2 

Calculat ional methods 2 

Program operation 5 

Program listing 11 

Program output 18 

References 19 



11 



SYMBOLS AND TERMINOLOGY 1 

e. = energy of the ith electronic level, cm - 
g* = degeneracy of the i th electronic level 

= symmetry number 

a = rotational vibrational interaction constant 
x = anharmonicity correction 
0) = fundamental frequency, cm - 

e 2 

1 = moment of interia, g cm 

S = entropy, cal mol~ K~ 

C = heat capacity, cal mol - K - 

H = enthalpy, cal mol A K 



r = bond distance, A 



e 



T = transition point temperature, K 

p -01 

h = Planck's constant, 6.62617636 x 10 z/ erg-sec 

k' = Boltzmann's constant, 1.38066244 x 10 erg molecule K 

R = gas constant, 1.98719.7 cal K~*mol~* 

R 1 = gas constant, 82.056826 atm cm 3 mol K 

N. = Avogadro's constant, 6.02204531 x 10 23 mol -1 

c = Speed of light, 2.9979245812 x 10 10 m sec -1 
M = molecular weight , g mol" 
e i = energy in ergs, (cm -1 x he) 



Values for the physical constant are from (2) . Underlined numbers in 
parentheses refer to items in the list of references at the end of this 
paper. 



A COMPUTER PROGRAM FOR CALCULATING THERMODYNAMIC 
PROPERTIES FROM SPECTROSCOPIC DATA 

by 

R. P. Beyer 2 



ABSTRACT 

A FORTRAN IV computer program has been written to calculate thermodyna- 
mic properties over a desired temperature range from spectroscopic data. 
This program computes heat capacities, enthalpies, and entropies for monatom- 
ic, diatomic, linear, and nonlinear polyatomic gases. This research is part 
of the Bureau of Mines effort to provide thermodynamic data for the advance- 
ment of mineral technology. 



2 

Chemical engineer, Albany Research Center, Bureau of Mines, Albany, Oreg. 



INTRODUCTION 

As part of a program to advance technology, reduce energy use, and abate 
pollution, the Bureau of Mines generates thermochemical data on various min- 
erals and metals. A computer program for calculating C~, S°(T)-S°(0), 
and H°(T)-H°(298) was written to further this aim. The program was written 
in FORTRAN IV Plus, Version 2.5, to run on a Digital Equipment Corp. 3 PDP 
11/34 minicomputer with the RSX-11M, Version 3.2, operating system. 
Although programs have been written previously for this type of calculation, 
[e.g., McBride and Gordon (5)], the vast differences in hardware, operating 
systems, and compilers necessitated a rewriting rather than an adaptation of 
previous work. 

The program accepts as input the electronic energy levels, degeneracies, 
moment of inertia, number of atoms, vibrational frequencies, vibrational- 
rotational interaction constant, symmetry number, and molecular weight. 
The output consists of the heat capacities, entropies, and relative enthal- 
pies over a given temperature range. The normal output consists of the ther- 
modynamic properties calculated over the desired temperature range at 
intervals starting at K and including 298.15 K. Any noninternal desired 
temperature may also be included . 

CALCULATIONAL METHODS 

The methods to calculate the thermodynamic properties follow the adapta- 
tion of Mayer and Mayer (4) by King and others (3) . For the definition of 
the symbols used, refer to the symbols and terminology section. 
(1) Monatomic gases 

Translational contribution: 

(C p ) tr = 5/2 R 



(H - H ). Y . = 5/2 RT 

tr 



S tr = RCJtnR^ 5/2 AnT + 3/2 &nM + 3/2 £n±™ - 5/2 AnN + 5/2) 



Reference to specific brand names does not imply endorsement by the Bureau 
of Mines. 



Electronic contribution: 



C e= R 



0^ - (Q^ 2 1 
. Q Q J 

[f] 



(H - H ) = RT H 



S e = R |AnQ + 



where 



v _e i /kT 
Q = I g ± e 



C 1 

e . 

1 



Q 1 = I 8i 

kT 
i l^J 



Q 11 = I 8l 



i L kT J 



•e 1 /kT 



2 -e./kT 

e 



(2) Diatomic gases 

Translational and electronic contributions: same as for monatomic 

gases. 

Rotational contribution 

c r = r(i + z£) 

r 45 

(H-H ) r = RT(l-^-^) 

v 2 
S, = R (1 - lny - lno - £-) 

r 90 



where y = 



hcB ( 
kT 



B = B - 1/2'acm -1 



B„ = *_ cm" 1 
e 8n z cl 



Vibrational contribution 



2 u 
r - R u e 
C vib " R (e u _ 1)2 



(« " H 0) v ib - 



RT 



(e u -l) 



S vib = R 



-S— - *n (l-e _u ) 
e u -l 



where u = 



hctoc 
kT 



u e e e' 
Anharmonicity corrections 



-1 



_ R 



cor 



4x 



u 3 e u (2ue u -2e u +u+2 ) 



(e u - 1)' 



+ 6 



u 3 e u (e u +l) 



(e u -l) 3 



+ 16Y 



< H " H o) C or " 



RT 



2x 



u 2 (2ue u - e u + 1 ) 



(e u -l) 3 



+ 6 



2 u 
u e 



+ 8Y 



(e u - 1) : 



cor 



4xu 2 e u + 6ue u f 6 + 16Y 



(e u -l)3 (e u -l) 2 (e u -l) u 



where x = 



l-2x. 



6 = 



Y = 



(3) Linear polyatomic gases 

Electronic: Same for all atoms and molecules. 

Trans lational: Same for all atoms and molecules. 

Rotational plus symmetry: Same as for diatomic molecules except 

for calculation of I(moraent of inertia) . 
Vibrational: Fundamental frequencies 3n-5 in number. Follow the 
diatomic method and sum the results. 

(4) Nonlinear polyatomic 

Electronic: Same as linear polyatomic. 
Translational: Same as linear polyatomic. 
Rotational plus symmetry: 

(H(T) - H(0)) r = (3/2)RT. 

C D = (3/2)R 
F r 

512u 7 k 3 , 
S r = 3/2R + R[l/2 In I A 1 B I C + 3/2 InT - Ino + 1/2 in g J 

Vibrational: Fundamental frequencies 3n-6 in number. Follow the 
diatomic method and sum the results. 

PROGRAM OPERATION 

Input consists of a data file containing the pertinent information. 
There are four different types of data input files, depending on which type 
of gas is being studied. All data in the examples are from JANAF (1). The 
following examples show the required format. 

(1) The monatomic input file: 

a) In general, 
Name 

Monatomic 

Highest temperature to be calculated 
K, the number of noninterval temperatures (Tp^, i = 1,2 K) 



T Pl 



Number of energy levels 



Electronic energy of level 1 (e.) 



F(g) 




Monatomic 




2000. 









12 







4 


404. 


2 


103327.14 


18 


115918.7 


6 


116597.23 


14 


117465.88 


22 


118627.73 


10 


123118.7 


12 


128346.36 


80 


132786.07 


16 


133531.81 


22 


134978.71 


88 


18.9984 





Degeneracy of 
level i (g ± ) 



Molecular weight 
b) Monatomic example, F(g): 



(2) The Diatomic input file: 

a) In general, 

Name 

Diatomic 

Highest temperature to be calculated 

Number of noninterval temperatures 



T Pi 



Number of energy levels 



Electronic energy of level i (e.) 



Degeneracy of 
level i (g i ) 



Atomic weight of atom 1 
Atomic weight of atom 2 
Internuclear distance (r in A) 
Fundamental frequency (u> in cm" ) 
Anharmonicity correction (uj x ) 

Rotational-vibrational interaction constant (a) 
Symmetry number (a) 

b) Diatomic example, CuF(g); 



CuF(g) 

Diatomic 

6000 



1 



63.54 

18.9984 

1.743 

621.89 

3.941 

0.004586 

1 



8 

(3) The linear polyatomic input file: 
a) In general, 



Name 

Linear polyatomic 

Highest temperature to be calculated 

Number of noninterval temperatures, (Tp) 



T Pi 



Number of energy levels 



Electronic energy of level i (e-) 



Degeneracy of 
level i (g ± ) 



Molecular weight 
Number of atoms 



Fundamental frequency of level i (u>. ), enter degenerate 

frequencies as many times as the degeneracy of the modes , 

see examples below. 

Moment of inertia (I) 

Rotational vibrational interaction constant (a) 

Symmetry number (a) 



b) Linear polyatomic example, CuF 2 (g): 

CuF2(g) 

Linear Polyatomic 

2000 



3 

2 

9000 4 

18000 4 

101.5368 

3 

608 

205 

205 

768 

112.4008735 (see below for units) 



2 

(4) The nonlinear polyatomic input file: 

a) In general, 

The same as the linear case except the second line should 
read nonlinear polyatomic and I is the product of the 
three principal moments of inertia I = I.Iglp. 

b) The nonlinear polyatomic example Zrl^g): 

ZrI4(g) 

Nonlinear polyatomic 

2000. 

2 

155.3 

623.9 

1 

1 

598.66 

5 

146. 

45. 

45. 
237. 
237. 
237. 

58. 

58. 

58. 
65718000 (see below for units) 

12 



10 



The FORTRAN-IV Plus compiler used with this program limits calculations 
to numbers between ±10 and ±10 . Therefore, values for the moment of 
inertia for linear and nonlinear polyatomics should be entered as follows: 

(1) Linear: values of I in units of g-cm should be multiplied by 
(N A /10 -16 ) before entering in the data input file. CuF 2 (g), I = 

18.7 x 10 g-cm . For the input file this number would be changed 
to (18.7 x 10 -39 )(N A /10 -16 ) = 112. A. 

(2) Polyatomic: Values for I.IgI~, which would have an exponent of 

10~ 117 [from (10~ 39 ) 3 ], should be entered without this exponent. 
For example, Zrl^Cg), I A I B Ic = 65718000 x 10" 117 . This should be 

entered as just 65718000. 

An example of the output file for the Zrl^Cg) example is given after the 
program listing. The program was checked by running all the examples shown 
and comparing the output with the accepted JANAF (1) results. Agreement 
between the computer output and the JANAF values was exact. 



11 



PROGRAM LISTING 



C 

C A PROGRAM TO CALCULATE HEAT CAPACITY, ENTHALPY, AND 

C ENTROPY FROM SPECTROSCOPIC DATA 

C 

Real*8 HT,e(100),g(100),w(100) ,xe(100) ,wexe(100) ,MW,MW1,MW2 

Real*8 re,a,sig,Be,BO,y,x,c,R,pi,k,h,kT,lny 

Real*8 c trans, htrans,s trans, crot,hrot,srot,cvlb,hvib,svib 

Real*8 celec , helec , selec , cr298 ,hr298 , sr298 ,ht298 , ct298 , st298 

Real*8 cv298,hv298,sv298,ce298,he298,se298 

Real*8 qO , ql , q2 ,MI , Rprime , Na , hout 

Real*8 del,hcor,scor,eu,htotal,h298,stotal,s298,eul,T(100) ,gam 

Real*8 Td,ccor,ctotal,c298,u,w0(100) 

Byte lgas,gasnam(30) 
c 

c Initialize 
c 

do 500 1=1,100 

e(i)=0.0 

T(i)=0.0 

g(i)=0.0 

w(i)=0.0 

wO(i)=0.0 

xe(i)=0.0 
500 wexe(i)=0.0 

Na=6.02204531e23 

Rprime =82. 056826 

k=1.38066244e-16 

h=6.62617636e-27 

c=2.9979245812el0 

R=l. 9871917 

pi=3. 141592654 
c 

c conversational input of constants 
c 

open( unit=l, name -'dll: gas.dat* ,type='new' .dispose"' save' ) 

write(5,130) 

130 format(/,lx, 'Enter name of the input file') 
read(5,131)(gasnam(i),i=5,30) 

131 format (30A1) 

type *, 'Output is in file gas.dat' 

gasnam(30)=O 

gasnam(l)='S' 

gasnam(2)='Y* 

gasnam(3)='0' 

gasnam(4)=' : ' 

open(unit=2,name=gasnam, type-' old' .dispose-'save' ) 

read(2,131)(gasnam(i),i=l,20) 

write(l,132)(gasnam(i),i=l,20) 

132 format (lx,20Al,/) 



12 



read(2,101)igas 
101 format(lAl) 

if(igas.eq.'M' .or.igas.eq. 'm' )go to 2 
if(igas.eq.'D' .or.igas.eq. 'd' )go to 3 
if (Igas.eq. 'L' .or.igas.eq. *1' )go to 4 
if (igas.eq. 'N' .or.igas.eq. 'n')go to 5 
type *,' INVALID GAS TYPE!' 
stop 

2 type * , ' monatomic gas * 
iflag=7 

go to 6 

3 type *,' diatomic gas' 
iflag=8 

go to 6 

4 type *,' linear polyatomic gas* 
iflag=9 

go to 6 

5 type *,' non-linear polyatomic gas' 
iflag=9 

go to 6 

6 read(2,*)HT 
iHT=HT 

ntemp=iHT/100 
T(l)=298.15 

do 43 i=l,ntemp 

43 T(i+l)=100.*i 
c 

c Read tranisition temperatures 
c 

read(2,*)nTp 

if(nTp.eq.0)go to 41 

do 42 i-l,nTp 
42 read(2,*)T(ntemp+l+i) 
c 

c Arrange the temperatures in ascending order 
c 

do 44 j=2,ntemp+nTp 

do 44 i»2,ntemp+nTp 

if(T(i).lt.T(i+l))go to 44 

Ttemp=<r(i) 

T(l)-T(i+1) 

T(i+1)-Ttemp 

44 continue 

41 read(2,*)nlevel 

do 50 i«l,nlevel 
50 read(2,*)e(i),g(i) 

If (igas.eq. 'M' .or. igas.eq. 'm' )go to 7 
if (igas.eq. 'D' .or.igas.eq. 'd')go to 8 
go to 9 
c 

c monatomic gas 
c 



read(2,*)MW 
go to 20 



13 
c 

c diatomic gas 
c 

8 read(2,*)MWl 
read(2,*)MW2 
read(2,*)re 

c 

c calculate moment of inertia 

c 

MI =( (MW1*MW2 ) / (MW1-HW2 ) )*re*re 

MW=MW14MW2 

read(2,*)w(l) 

read(2,*)wexe(l) 

read(2,*)a 

read(2,*)sig 

nvib=4 

go to 20 
c 

c polyatomic gas 
c 

9 read(2,*)MW 
read(2,*)natom 
nvib=3*natom-6 

if (igas .eq. 'L' .or.igas.eq. '1' )nvib=3*natom-5 

do 51 i=l,nvib 
51 read(2,*)w(i) 

read(2,*)MI 

read(2,*)a 

read(2,*)sig 

iharm=0 
c 

20 write(5,300) 
300 format(//,6x, , T , ,10x, , Cp , ,10x, , H-H298\llx, , S , ) 

write(l,300) 

write(l,302) 
302 format(6x, 'K' ,7x»'cal/mol-K' ,6x, 'kcal/mol' ,5x, 'cal/mol-K' ,/) 
c 

c start calculations 
c 

ht298«O.0 

hr298^).0 

he298«O.0 

hv298*).0 

hc298-O.0 

istart-1 
17 do 55 i-i8tart,ntemp+nTp+l 

if (igas .eq. 'M' .or.igas.eq. 'm' )go to 16 
c 

c rotational calulations 
c 

12 if(igas.eq. 'D* .or.igas.eq. 'd' )go to 13 

if (igas .eq . 'L' .or .igas .eq. ' 1' )go to 13 
c 
c non-linear polyatomic rotational 



14 

c 

crot=1.5*R 

hrot=T(i)*crot+hr298 

srot=(R/2.)*(log(512.*pi**7*((k*l.el6)**3)/((h*l.e27)**6)) 
&-3.*log(10.)+3.)+(R/2.)*(log(MI)+3.*log(T(i))-2.*log(sig)) 

go to 14 
c 

c diatomic or linear polyatomic rotational 
c 

13 Be=((n*Na)/(8.*pi*pi*c*l.e-16))/MI 
B0=$e-0.5*a 
y=(((n*Na)*h/(k*8.*pi*pi*l.e-16))/(T(i)*Ml)) 

&-((h*c)/(2.*k))*(a/T(i)) 

crot=R*(l .+(y*y/45.)) 

hrot=R*T(i)*(l.-y/3.-(y*y/45.))+hr298 

lny=O.0 

if(y.gt.O.O)lny=log(y) 

srot=R*(l.-lny-log(sig)-(y*y/90.)) 
c 

c vibrational calculations 
c 

14 cvib=0.0 
hvib=O.0 
svib=O.0 
ccor=O.0 
hcor=0.0 
scor=O.0 

do 53 j=l,nvib 

w0(j)=v(j)-2.*wexe(j) 

u=((h*c)/k)*wO(j)/T(i) 

eu=exp(u) 

cvib=(R*u*u*eu/ ( (eu-1 . )*(eu-l . ) ) )+cvib 

hvib=(R*T(i)*u/(eu-l . ) )+hvib 

svib=(R*( (u/ (eu-1 . ) )-log(l .-exp(-u) ) ) )+svib 
c 

c anharmonicity corrections 
c 

if(nvil.eq.l)go to 15 

if(iharm.ne.l)go to 53 

15 xe(j)-wexe(j)/w(j) 
del-0.0 

if(a.ne.0.0)del«a/(Be-a/2.) 
gam=Be/w(j) 

eul "eu-1 . 

x=xe(j)/(l.-2.*xe(j)) 

ccor-(R/u)*((4.*x*u*u*u*eu*(2.*u*eu-2.*eu+u+2.)/(eul*eul*eul*eul)) 
&+( del*u*u*u*eu*( eu+1 . ) / ( eul*eul*eul ) )+16 . *gam)+ccor 

hcor-(R*T(i)/u)*(((2.*x)*u*u*(2.*u*eu-eu+l.)/(eul*eul*eul))+ 
&( de l*u*u*eu/ ( eul*eul ) )+8 . *gam)+hcor 

scor«**((4.*x*u*u*eu/(eul*eul*eul))+(del*u*eu/(eul*eul))+(del/eul) 
&+(16.*gam/u))+scor 
53 continue 

hvib=hvib+hv298 

hcor=hcor+hc298 



15 
c 

c Translational calculations 
c 

16 ctrans«2.5*R 

htrans=ctrans*T(i)+ht298 

strans=R*(2.5*log(T(i))+1.5*log(MW) 
&+log(Rprime)+1.5*log(2.*pi*k)-3*log(h)-2.5*log(Na)+2.5) 
c 

c electronic calculations 
c 

celec=0.0 

helec=O.0 

selec=O.0 

call Q(e,g,nlevel,T(i),qO,ql,q2) 

if(q0.eq.O)go to 22 

celec=R*((q2/qO)-(ql/qO)*(ql/qO)) 

helec=**T(i)*ql/qO+he298 

selec=R*(log(qO)+(ql/qO)) 
c 

c calculate total thermodynamic functions 
c 

22 ctotal=ctrans+celec+crot+cvib+ccor 

htotal=htrans+hrot+hvib+helec+hcor 

stotal=strans+srot+svib+selec+scor 
c 

c if T=298.15 then go print values 
c 

if(i.eq.l)go to 18 

jflag=0 

if(T(i).eq.200)jflag=l 

write( 5, 122 )T(i),c trans, htrans.s trans 

122 format(/,lx,F9.2,3(lx,fl2.4)) 
vrite(5,123)celec,helec,selec 

123 format(10x,3(lx,fl2.4)) 

if (iga8.eq.'M' .or.igas.eq. 'm')go to 21 
write(5,123)crot,hrot,srot 
write(5,123)cvib,hvib,svib 
write(5,123)ccor,hcor,scor 
21 vrite(5,123)ctotal,htotal,stotal 
hout-htotal/1000 . 

write(l,301)T(i),ctotal,hout,stotal 
301 format(lx,f8.2,3(3x,fll.3)) 
if( jflag.eq.0)go to 55 
T(i)-298.15 
ccor«cc298 
hcor«hc298 
scor-8c298 
crot-cr298 
hrot-0.0 
srot-sr298 
ctrans«ct298 
htrans«=0.0 
8trans-st298 
cvib«cv298 



16 

hvib=O.0 
svib=sv298 
celec=ce298 
helec=O.0 
selec=se298 
hcor=0.0 
go to 22 
55 continue 
go to 19 

18 hc2~8=>-htrans 
hr298=-hroc 
he298~-*-hel^ 
hv298=-hvib 
hc298=-hcor 
h298=-htotal 
cr298=crot 
ct298=ctrans 
cv298=cvib 
cc298=ccor 
ce298^celec 
sr298=srot 
st298=strans 
sv298=svib 
sc298=scor 
se298=selec 
c298=ctotal 
s298=stotal 
write(5,124)ht298 

124 format(5x, '0' ,13x, '0' ,4x,f 12.4,10x, *0' ,5x, "Trans lational' ) 
write(5,125)he298 

125 format(19x, '0' ,4x,f 12.4,10x, '0' ,5x, 'Electronic' ) 
if (igas.eq.'M' .or.igas.eq. 'm' )go to 25 
write(5,200)hr298 

200 format(19x, '0' ,4x,f 12.4,10x, '0' ,5x, 'Rotational' ) 
write(5,201)hv298 

201 format(19x, '0' ,4x,f 12.4,10x, '0' ,5x, 'Vibrational' ) 
write(5,202)hc298 

202 format(19x, '0' ,4x,f 12.4,10x, '0' ,5x, 'Anharmonic' ) 
25 write(5,203)h298 

203 format(19x, , 0•,4x,fl2.4,10x,•0•,5x,•Total , ) 
hout«=h298/1000. 

write(l,204)hout 

204 format(8x,'0',13x,'0',3x,fll.3,13x,'0') 
is tart -2 

go to 17 

19 continue 
close (unit-1) 
end 

c 

c subroutine to calculate partition functions 

c 

subroutine Q(e,g,nlevel,T,q0,ql,q2) 

Real*8 e(nlevel) ,g(nlevel) ,T,q0,ql,q2,kT,ekT,ge 

kT=1.438785935/T 



17 
qO=-0.0 
ql=0.0 
q2=O.0 

do 1 i=l,nlevel 
ekT=e(i)*kT 
ge=exp(-ekT)*g(i) 
qO=qO+ge 
ql=ql+ekT*ge 
q2=q2+ekT*ekT*ge 
return 
end 



18 



PROGRAM OUTPUT 



Zrl4(g) 








T 


Cp 


H-H298 


S 


K 


cal/mol-K 


kcal/mol 


cal/mol-K 








-6.330 





100.00 


21.258 


-4.743 


80.982 


200.00 


24.284 


-2.430 


96.897 


298.15 


25.090 


0.000 


106.772 


300.00 


25.098 


0.046 


106.927 


400.00 


25.410 


2.574 


114.196 


500.00 


25.559 


5.123 


119.884 


600.00 


25.642 


7.684 


124.552 


700.00 


25.692 


10.251 


128.509 


800.00 


25.725 


12.822 


131.942 


900.00 


25.748 


15.395 


134.973 


1000.00 


25.764 


17.971 


137.687 


1100.00 


25.776 


20.548 


140.143 


1200.00 


25.785 


23.126 


142.386 


1300.00 


25.792 


25.705 


144.450 


1400.00 


25.798 


28.284 


146.362 


1500.00 


25.802 


30.864 


148.142 


1600.00 


25.806 


33.445 


149.807 


1700.00 


25.809 


36.026 


151.372 


1800.00 


25.812 


38.607 


152.847 


1900.00 


25.814 


41.188 


154.243 


2000.00 


25.816 


43.769 


155.567 



19 



REFERENCES 

1. Dow Chemical Co. Thermal Research Laboratory. JANAF Thermochemical 

Tables, 2d ed., NSRDS-NBS 37, S/N 03030872, U.S. Government Printing 
Office, Washington, D.C., 1971, 1141 pp. 

2. International Union of Pure and Applied Chemistry, Division of Physical 

Chemistry, Commission of Physicochemical Symbols, Terminology, and 
Units. Manual of Symbols and Terminology for Physicochemical Quantities 
and Units. 1979, p. 35. 

3. King, E. G., A. D. Mah, and L. B. Pankratz. Thermodynamic Properties of 

Copper and Its Inorganic Compounds, INCRA Monograph Series II (sponsored 
by The International Copper Research Association and the U.S. Bureau of 
Mines), New York, 1973, pp. 5-8. 

4. Mayer, J. E., and M. G. Mayer. Statistical Mechanics. John Wiley and 

Sons, Inc., London, 1940, 495 pp. 

5. McBride, B. J., and S. Gordon. Fortran IV Program for Calculation of 

Thermodynamic Data. NASA TND-4097, 1967, 129 pp. 



trU.S GOVERNMENT PRINTING OFFICE: 1981-505-002/131 int.-bu.of mines,pgh.,pa. 25902 



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